How to find groups? (and how to use them in Erdős geometry?)

Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ∈ ℝ[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be re-written as z = f(x, y), for a polynomial or rational function f ∈ ℝ(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on “circle grids”).The main result of our paper concerns low-degree algebraic sets F which contain “too many” points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).